Behaviour index fluids

In equation 3.51, p is the density and n the power law flow behaviour index of the fluid. 51 and S2 are the cross-sectional areas of the smaller and larger pipes respectively and u the volumetric average velocity in the smaller pipe. 

The simple concept of an average mixer shear rate has been widely used in laboratory and industrial work and in most applications it has been assumed that the shear rate constant, k, is only a function of impeller type. Research is continuing on the possible influence of flow behaviour index and elastic properties, and also on procedures necessary to describe power consumption for dilatant fluids. It should be noted that in all aspects of power prediction and data analysis, power law models (equation 8.12) should only be used with caution. Apparent variability of k, may be due to inappropriate use of power law equations when calculations are made it should be ascertained that the average shear rates of interest (y = k N) lie within the range of the power law viscometric data. 

In these equations, m and n are two empirical curve-fitting parameters and are known as the fluid consistency coefficient and the flow behaviour index respectively. For a shear-thinning fluid, the index may have any value between 0 and 1. The smaller the value of n, the greater is the degree of shear-thinning. For a shear-thickening fluid, the index n will be greater than unity. When n = 1, equations (1.12) and (1.13) reduce to equation (1.1) which describes Newtonian fluid behaviour. 

This suggests that any correction factor which will cause the holdup data for shear-thinning fluids to collapse onto the Newtonian curve, must become progressively smaller as the liquid velocity increases and the flow behaviour index, n, decreases. Based on such intuitive and heuristic considerations, Farooqi and Richardson [1982] proposed a correction factor, J, to be applied to the Lockhart-Martinelli parameter, x, so that a modified parameter Xmod is defined as ... 

Re = pVl x"jm where jc is the distance (from the leading edge) at which the flow ceases to be streamline. For Newtonian fluids (n = 1), B(n) = 0.323 and RCc = 10 which is the value at the transition point. For a Uquid of flow behaviour index, n = 0.5, the limiting value of the Reynolds mnnber is 1.14 X lO which is an order of magnitude smaller than the value for a Newtonian fluid. 

The rheological behaviour of a china clay suspension of density 1200kg/m is well approximated by the Herschel-BuDdey fluid model with consistency coeflicientof 11.7Pa-s", flow behaviour index of 0.4 and yield stress of 4.6 Pa. Estimate the terminal fafling velocity of a steel ball, 5 mm diameter and density 7800kg/m. What is the smallest steel ball which will just settle under its own weight in this suspension ... 

A coal-in-oil slurry which behaves as a power-law fluid is to be heated in a double-pipe heat exchanger with steam condensing on the annulus side. The inlet and outlet bulk temperatures of the slurry are 291 K and 308 K respectively. The heating section (inner copper tube of 40 mm inside diameter) is 3 m long and is preceded by a section sufficiently long for the velocity profile to be fully estabhshed. The flow rate of the slurry is 400kg/h and its thermo-physical properties are as follows density = 900 kg/m heat capacity = 2800 J/kg K thermal conductivity = 0.75 W/mK. In the temperature interval 293 < T < 368 K, the flow behaviour index is nearly constant and is equal to 0.52. 

Figure 2.21 Inverse superficial filtrate rate to the power of the fluid behaviour index plotted volume of filtiate per unit area [Murase et al, 19S9a]...

For slurries exhibiting power-law fluid rheology, the transition velocity from laminar to turbulent flow is governed by the flow behaviour index n of the slurry. The equation proposed by Hanks and Ricks (1974) gives an estimate of this transition velocity in terms of the generalized Reynolds number Re. ... 

In turbulent flow, the Fanning friction factor ff for the slurry described by a power-law fluid model depends on both the generalized Reynolds number Re and the flow behaviour index n. 

In these equations K is fluid consistency, n is flow behaviour index, Xy is yield shear stress, Xv, is wall shear stress, D = 2 is the pipe diameter. Equation (4) may be rewritten for the friction factor... 

The value of fluid behaviour index n considerably affects both turbulent models, which can relatively well approximate turbulent slurry flow if the value of n is correctly predeterminate. 

The flow behaviour index n decreases and values of both fluid consistency K and yield shear stress increase with increasing slurry concentration Cy. 

B roughness function c concentration d particle diameter D pipe diameter / friction factor i pressure gradient K fluid consistency n fluid behaviour index r cylindrical coordinate R pipe radius Re Reynolds number u local velocity Re Reynolds number V mean velocity V, friction velocity K area ratio a Karman s constant shear stress ratio T shear stress Ty yield shear stress p density... 

Show how. by suitable selection of the index n, the power law may be used to describe the behaviour of both shear-thinning and shear-thickening non-Newtonian fluids over a limited range of shear rates. What are the main objections to the use of the power law Give some examples of different types of shear-thinning fluids. 

A fluid which exhibits non-Newtonian behaviour is flowing in a pipe of diameter 70 mm and the pressure drop over a 2 m length of pipe is 4 x 104 N/m2. A pitot lube is used to measure the velocity profile over the cross-section. Confirm that the information given below is consistent with the laminar flow of a power-law fluid. Calculate the power-law index n and consistency coefficient K. 

There is an expression that does not truly fit either class of behaviour, for power law fluids which can be expressed in terms of stress, rate or apparent viscosity with relative ease. They can describe shear thickening or thinning depending upon the sign of the power law index n ... 

In Figure 3.28, the shear stress is shown as a function of shear rate for a typical shear-thinning fluid, using logarithmic coordinates. Over the shear rate range (ca 10 to lO-" s the fluid behaviour is described by the power-law equation with an index n of 0.6, that is the line CD has a slope of 0.6. If the power-law were followed at all shear rates, the extrapolated line CCDD would be applicable. Figure 3,29 shows the corresponding values of apparent viscosity and the line CCDD has a slope of n — I = —0.4. It is seen that it extrapolates to jXa = oo at zero shear rate and to /Zq = 0 at infinite shear rate. 

It is now established both theoretically and experimentally that many thermodynamic variables assume a simple power-law behaviour at or near critical points in both pure and mixed fluids. The actual functional dependence of one variable on another can be characterized by the so-called critical indices a, 5, etc. The critical index j8, for example, defines both the shape of the gas-liquid coexistence curve for a pure fluid and the liquid-liquid coexistence curve of a binary mixture in the vicinity of either an upper or a lower critical solution temperature. The correspondence between critical phenomena in one-, two-,... 

Thus, the index m is the slope of the log-log plots of the wall shear stress Xw versus (8V/D) in the laminar region (the limiting condition for laminar flow is discussed in Section 3.3). Plots of x versus (8V/D) thus describe the flow behaviour of time-independent non-Newtonian fluids and may be used directly for scale-up or process design calculations. 

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